Further Consequences of the Colorful Helly Hypothesis

Let F be a family of convex sets in R^d, which are colored with d+ 1 colors. We say that F satisfies the Colorful Helly Property if every rainbow selection of d+ 1 sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovász states that for any such colorful family F there is a color class F_i⊂ F, for 1 ≤ i≤ d+ 1 , whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension d≥ 2 there exist numbers f(d) and g(d) with the following property: either one can find an additional color class whose sets can be pierced by f(d) points, or all the sets in F can be crossed by g(d) lines.